January  2013, 12(1): 519-546. doi: 10.3934/cpaa.2013.12.519

Numerical study of a family of dissipative KdV equations

1. 

LAMFA, UMR 6140, Université de Picardie Jules Verne, Pôle Scientifique, 33, rue Saint Leu, 80039 Amiens, France, France

Received  April 2011 Revised  December 2011 Published  September 2012

The weak damped and forced Korteweg-de Vries (KdV) equation on the 1d Torus have been analyzed by Ghidaglia[8, 9], Goubet[10, 11], Rosa and Cabral [3] where asymptotic regularization e ects have been proven and observed numerically. In this work, we consider a family of dampings that can be even weaker, particularly it can dissipate very few the high frequencies. We give numerical evidences that point out dissipation of energy, regularization e ect and the presence of special solutions that characterize a non trivial dynamics (steady states, time periodic solutions).
Citation: Jean-Paul Chehab, Georges Sadaka. Numerical study of a family of dissipative KdV equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 519-546. doi: 10.3934/cpaa.2013.12.519
References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

show all references

References:
[1]

M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete schrodinger equations and dissipative dynamical systems,, Communications on Pure and Applied Analysis, 7 (2008), 211.   Google Scholar

[2]

M. Abounouh, H. Al Moatassime, C. Calgaro and J-P. Chehab, A numerical scheme for the long time simulation of a forced damped KdV equation,, in preparation., ().   Google Scholar

[3]

M. Cabral and R. Rosa, Chaos for a damped and forced KdV equation,, Phys. D, 192 (2004), 265.  doi: 10.1016/j.physd.2004.01.023.  Google Scholar

[4]

C. Calgaro, J.-P. Chehab, J. Laminie and E. Zahrouni, Schémas multiniveaux pour les équations d'ondes,, (French) [Multilevel schemes for waves equations], 27 (2009), 180.  doi: 10.1051/proc/2009027.  Google Scholar

[5]

J.-P. Chehab and B. Costa, Time explicit schemes and spatial finite differences splittings,, Journal of Scientific Computing, 20 (2004), 159.   Google Scholar

[6]

J.-P. Chehab and G. Sadaka, Numerical study of a family of dissipative KdV equations,, preprint, (2011).   Google Scholar

[7]

D. Dutykh, Modélisation mathématique des Tsunamis,, (French) [Mathematical modeling of Tsunamis], (2007).   Google Scholar

[8]

J-M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical system in the long time,, J. Diff. Eq., 74 (1988), 369.  doi: 10.1016/0022-0396(88)90010-1.  Google Scholar

[9]

J-M. Ghidaglia, A note on the strong convergence towards attractors for damped forced KdV equations,, J. Diff. Eq., 110 (1994), 356.  doi: 10.1006/jdeq.1994.1071.  Google Scholar

[10]

O. Goubet, Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations,, Discrete Contin. Dynam. Systems, 6 (2000), 625.   Google Scholar

[11]

O. Goubet and R. Rosa, Asymptotic smoothing and the global attractor of a weakly damped KdV equation on the real line,, J. Differential Equations, 185 (2002), 25.  doi: 10.1006/jdeq.2001.4163.  Google Scholar

[12]

D. Gottlieb and S. A. Orszag, "Numerical Analysis of Spectral Methods: Theory and Applications,", SIAM Philadelphia, (1977).  doi: 10.1115/1.3424477.  Google Scholar

[13]

S. Mallat, "A Wavelet Tour of Signal Processing,", Academic press, (1998).   Google Scholar

[14]

A. Miranville and R. Temam, "Mathematical Modeling in Continuum Mechanics,", Cambridge University Press, (2005).   Google Scholar

[15]

L. Molinet and S. Vento, Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case,, preprint, ().   Google Scholar

[16]

E. Ott and R. N. Sudan, Nonlinear theory of ion acoustic wave with Landau damping,, Physics of fluids, 12 (1969), 2388.  doi: 10.1063/1.1692358.  Google Scholar

[17]

E. Ott and R. N. Sudan, Damping of solitary waves,, Physics of fluids, 13 (1970), 1432.  doi: 10.1063/1.1693097.  Google Scholar

[18]

A. Duràn and J. M. Sanz-Serna, The numerical integration of a relative equilibrium solutions. the nonlinear schrodinger equation,, IMA J. Num. Anal., 20 (2000), 235.  doi: 10.1093/imanum/20.2.235.  Google Scholar

[19]

R. Temam, "Infinite Dimensional Dynamical Systems in Mechanics and Physics,", 2$^{nd}$ edition, (1997).  doi: 10.1007/978-1-4684-0313-8.  Google Scholar

[20]

S. Vento, Global well-posedness for dissipative Korteweg-de Vries equations,, Funkcialaj Ekvacioj, 54 (2011), 119.  doi: 10.1619/fesi.54.119.  Google Scholar

[21]

S. Vento, Asymptotic behavior for dissipative Korteweg-de Vrie equations,, Asymptot. Anal., 68 (2010), 155.   Google Scholar

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